When I stated that the middle is roughly exponential, this was the graph that I was looking at:

d <- density(karma)

plot(log(d$y) ~ d$x)

I don't do this for a living, so I am not sure at all, but if I really really had to make this formal, I would probably use maximum likelihood to fit an exponential distribution on the relevant interval, and then Kolmogorov-Smirnoff. It's what shminux said, except there is probably no closed formula because the cutoffs complicate the thing. And at least one of the cutoffs is really necessary, because below 3 it is obviously not exponential.

I expected something like this or the section thereafter.